System and method for performing risk analysis

ABSTRACT

A computerized data processing system for performing risk analysis of a portfolio, the system including a modeling and calibration unit configured to describe risk factors as random variables, the random variables being related to each other by a correlation matrix; an input unit configured to enter or choose calibration data and to obtain, by using the modeling and calibration unit, values for parameters that describe the degree of freedom for sub-vectors and to obtain values for the correlation matrix for the random variables, to enter or choose at least one risk mapping function, and to enter portfolio data of a portfolio to be analyzed; a simulation unit configured to simulate realization of the risk factors by using the correlation matrix; and an output unit configured to generate output data resulting from the simulation unit in a form of at least one of a risk measure or a price.

BACKGROUND OF THE INVENTION

The present invention relates to a system and a computer implementedmethod for performing risk analysis of a portfolio.

The financial services industry, especially the financial riskmanagement departments and the financial security pricing departments ofinsurance and re-insurance companies and banks, has established in thepast tools and means for estimating their financial risk. Such risks canbe associated with credit instruments and portfolios of creditinstruments, such as bonds and loans. Such risks can also be associatedwith equity portfolios of various currencies or insurance andreinsurance liabilities.

These tools and means are based on models, based on which simulationsare performed to generate possible valuation scenarios. Thesesimulations generally use the Monte Carlo method or other appropriatemethods. The models use probability distributions and are calibratedwith historical data. Such historical data may be obtained from varioussources, such as DataStream™.

These simulations are usually implemented in computer software as partof a financial services system and are run on computer hardware.

The input data for the simulations are risk factors, which are handledas random variables. Such risk factors can be equity indices, foreignexchange rates, interest rates, or insurance loss frequencies andseverities. The result or output data of such simulations is at leastone risk measure in the form of a numerical quantity or value. Usually,several risk measure values of different types can be obtained.

These risk measure values will be forwarded to an analyst or an actuaryor an underwriter, i.e. a human representative of a financial servicescompany. These risk measure values enable him to decide whether or notany actions should be taken to reduce the risk. Such actions can bechanges in a credit or equities portfolio, or in a portfolio ofinsurance and reinsurance liabilities.

The risk measures usually consist of a variety of values, such as themaximum value obtained, the standard deviation of the simulation, ashortfall, usually the 99% shortfall, or a value-at-risk (VAR™). TheVAR™ is the greatest possible loss that the company may expect in theportfolio in question with a certain given degree of probability duringa certain future period of time. The full distribution itself can be therisk measure as well.

Typically a large number of risk factors have to be considered.Therefore, multidimensional probability distributions have to be used.As the risk measures are often determined at the tail of suchdistributions, a precise modelling of the tail dependency is important.

Furthermore, the dependency of the risk factors has to be considered.However, when using a linear correlation, the dependency is often notmodelled adequately. One known solution to better model dependency isthe use of copulas.

These copulas are well known in the state of the art. They are jointdistribution functions of random vectors with standard uniform marginaldistributions. They provide a way of understanding how marginaldistributions of single risks are coupled together to form jointdistributions of groups of risks.

Different kinds of copulas are known. Examples of closed form copulasare the Gumbel and the Clayton copula. Examples of implicit copulas,i.e. copulas for which no closed form exists, are the Gaussian copulaand the t-copula.

It has become increasingly popular to model vectors of risk factor logreturns with so-called meta-t distributions, i.e. distributions with at-copula and arbitrary marginal distributions. The reason for this isthe ability of the t-copula to model the extremal dependence of the riskfactors and also the ease with which the parameters of the t-copula canbe estimated from data. In Frey Rüdiger et al., “copulas and creditmodels” RISK, October 2001, p.p. 111-114, the use of such t-copulas formodelling credit portfolio losses is described. The disclosure thereofis herein implemented by reference.

We will recall therefore only the basic definitions and properties oft-distributions and t-copulas. For more on copulas in general, seeNELSEN, R. (1999): An Introduction to copulas. Springer, New York, orEMBRECHTS, P., A. MCNEIL, AND D. STRATTON (2002): “Correlation andDependence in Risk Management: Properties and Pitfalls,” in RiskManagement: Value at Risk and Beyond, ed. By M. Dempster, pp. 176-223.Cambridge University Press, Cambridge.

Before describing the state of the art and the present invention ingreater detail it is helpful to define the various variables and values.The following notation is used in the description of the prior art aswell as of the invention:

-   d dimension, number of risk factors-   R^(d) the d-dimensional usual real vector space-   E(X) expected value of the random variable X-   Var(X) variance of the random variable X-   Cov (X,Y) covariance of the random variables X and Y-   X,Y,Z random vectors-   Cov(X) covariance matrix of X-   ν number of degrees of freedom-   Σ covariance matrix-   N_(d)(0,Σ) d-dimensional Gaussian distribution with mean 0 and    covariance Σ-   φ univariate Gaussian distribution function-   χν² Chi Square distribution with degree of freedom ν-   ρ correlation matrix-   t_(ν) Student's t distribution function with degree of freedom ν-   t_(ν) ⁻¹ Student's t quantile function-   t_(ν,ρ) ^(d) Student's t d-dimensional distribution function with    correlation matrix ρ and degree of freedom ν-   Γ usual gamma function-   det A determinant of matrix A-   H_(k) arbitrary univariate distribution function-   U random variable uniformly distributed on [0,1]-   τ(X,Y) Kendall's tau rank correlation for random variables X and Y-   a_(k) credit multi-factor model parameters-   P└A┘ probability of occurrence of event A P└X≦x┘ probability that X    is lower or equal than x-   λ_(k) counterparty idiosyncratic parameter-   E_(k) credit exposure on counterparty k-   l_(k) loss given default

Let Z˜N_(d)(0,Σ) and U (random variable uniformly distributed on [0,1])be independent. Furthermore, G denotes the distribution function of√{square root over (ν|x_(ν) ²)} and R=G⁻¹(U).

Then the R_(d)—valued random vector Y given byY=(RZ ₁ ,RZ ₂ ,RZ ₃ , . . . ,RZ _(d))′  (1)has a centered t-distribution with ν degrees of freedom. Note that forν>2,

${{Cov}(Y)} = {\frac{v}{v - 2}{\Sigma.}}$By Sklar's Theorem, the copula of Y can be written asC _(ν,ρ) ^(t),(u)=t _(ν,ρ) ^(d)(t _(ν) ⁻¹(u ₁), . . . ,t _(ν) ⁻¹(u_(d))),  (2)where ρ_(i,j)=Σ_(ij)/√{square root over (Σ_(ii)Σ_(jj))} for i, jε{1, . .. , d} and where t_(ν,ρ) ^(d) denotes the distribution function of√{square root over (ν)}Z/√{square root over (S)}, where S˜x_(ν) ² andZ˜N_(d)(0,ρ) are independent (i.e. the usual multivariate t distributionfunction) and t_(ν) denotes the marginal distribution function oft_(νρ,) ^(d) (i.e. the usual univariate t distribution function). In thebivariate case the copula expression can be written as

$\begin{matrix}{{C_{v,\rho}^{t}\left( {u,v} \right)} = {\int_{- \infty}^{t_{v}^{{- 1}{(u)}}}{\int_{- \infty}^{t_{v}^{{- 1}{(v)}}}{\frac{1}{2{\pi\left( {1 - \rho_{12}^{2}} \right)}^{1/2}}\left\{ {1 + \frac{s^{2} - {2\rho_{12}{st}} + t^{2}}{v\left( {1 - \rho_{12}^{2}} \right)}} \right\}^{{- {({v + 2})}}/2}\ {\mathbb{d}s}\ {{\mathbb{d}t}.}}}}} & (3)\end{matrix}$

Note that ρ₁₂ is simply the usual linear correlation coefficient of thecorresponding bivariate t_(ν)-distribution if ν>2. The density functionof the t-copula is given by

$\begin{matrix}{{{c_{v,\rho}^{t}\left( {u_{1},\ldots\mspace{11mu},u_{d}} \right)} = {\frac{1}{\sqrt{\det\;\rho}}{\frac{{\Gamma\left( \frac{v + d}{2} \right)}{\Gamma\left( \frac{v}{2} \right)}^{d - 1}}{{\Gamma\left( \frac{v + 1}{2} \right)}^{d}} \cdot \frac{\prod\limits_{k = 1}^{d}\left( {1 + \frac{y_{k}^{2}}{v}} \right)^{\frac{v + 1}{2}}}{\left( {1 + \frac{y^{\prime}\rho^{- 1}y}{v}} \right)^{\frac{v + d}{2}}}}}},} & (4)\end{matrix}$where y_(k)=t_(ν) ⁻¹(u_(k)).

Let H₁, . . . , H_(d) be arbitrary continuous, strictly increasingdistribution functions and let Y be given by (1) with Σ a linearcorrelation matrix. Thenx=(H ₁ ⁻¹(t _(ν)(Y ₁)), . . . ,H _(d) ⁻¹(t _(ν)(Y _(d)))′  (5)has a t_(ν)-copula and marginal distributions H₁, . . . , H_(d). Thedistribution of X is referred to as a meta-t distribution. Note that Xhas a t-distribution if and only if H₁, . . . , H_(d). are univariatet_(ν)-distribution functions.

The coefficient of tail dependence expresses the limiting conditionalprobability of joint quantile exceedences. The t-copula has upper andlower tail dependence with ( t _(ν+1)(x)=1−t_(ν+1)(x)):λ=2 t _(ν+1)(√{square root over (ν+1)}√{square root over(1−ρ₁₂)}/√{square root over (1+ρ₁₂)})<0,in contrast with the Gaussian copula which has λ=0. From the aboveexpression it is also seen that the coefficient of tail dependence isincreasing in ρ₁₂ and, as one would expect since a t-distributionconverges to a normal distribution as ν tends to infinity, decreasing inν. Furthermore, the coefficient of upper (lower) tail dependence tendsto zero as the number of degrees of freedom tends to infinity for ρ₁₂<1.

The calibration of the copula parameters (ρ,ν) are typically done asfollows:

(i) Kendall's tau ρ(X_(i),Y_(j)) is estimated for every pair of riskfactor log returns. An estimate of the parameter ρ in (2) is obtainedfrom the relation

$\begin{matrix}{{\tau\left( {X_{i},Y_{j}} \right)} = {\frac{2}{\pi}{\arcsin\left( \rho_{ij} \right)}}} & (6)\end{matrix}$which holds for any distribution with strictly increasing marginaldistribution functions and a copula of an elliptical distribution whichhas a density, i.e. essentially any meta-elliptical distribution onewould consider in applications. Note that in high-dimensionalapplications an estimate of obtained from (6) may have to be modified toassure positive definiteness. This can be done by applying the so-calledeigenvalue method, i.e. the negative eigenvalues are replaced by a smallpositive number. Other calibrations are possible too.

(ii) Transforming each log return observation X_(i) with its respectivedistribution function, e.g. gaussian N₁(0,σ_(i)) yields, under themeta-t assumption, a sample from a t-copula with known ρ-parameter.Finally, the degrees of freedom parameter ν is estimated by standardmaximum likelihood estimation using (4).

In step (ii) the empirical marginals or fitted distribution functionsfrom a parametric family can be used.

The simulation from t-copula comprises the following steps:

(i) Draw independently a random variate Z from the d-dimensional normaldistribution with zero mean, unit variances and linear correlationmatrix ρ, and a random variate U from the uniform distribution on (0,1).

(ii) Obtain R by setting R≈G_(ν) ⁻¹(U). By (1) we obtain a randomvariate Y from the t-distribution.

(iii) Finally,(t_(ν)(Y₁), . . . ,t_(ν)(Y_(d)))′is a random variate from the t-copula

This meta-t assumption makes sense for risk factors of similar type,e.g. foreign exchange rates. However, it was found that it does notaccurately describe the dependence structure for a set of risk factorlog returns where the risk factors are of very different type, forexample a mixture of stock indices, foreign exchange rates and interestrates.

It is a general problem of such models, that the number of availablehistorical data is quite small, so that at least the tail dependency canhardly be modelled. Similar problems are also known in other fields, forexample in the combination reinsurance portfolios, the reliability ofindustrial complexes or in the weather forecast.

SUMMARY OF THE INVENTION

It is therefore a technical object of the invention to provide a systemand a computer implemented method for performing risk analysis bycombining a multiple of interdependent risk factors, wherein realhistorical data are used for calibration of a model, the model beingused as basis for simulations for predicting the present or the future,and wherein at least one risk measure is obtained which describes anactual or a future risk or a price is obtained. The inventive method andthe method shall be more flexible and accurate than the known systembased on the meta-t-model, but without the need to use more efficientdata processing machines and without the need to have an increasednumber of input data based on historical data.

This is achieved by a system and a method according to claim 1 and 6,respectively.

The invention still uses t-copulas. However, in the inventive system andmethod, the financial risk factors, i.e. the random variables, aregrouped into groups of different types and each group obtains its owndegree-of-freedom parameter. Therefore, a random vector can be obtainedwhich is partitioned into subvectors. Each subvector is properlydescribed by a multi-dimensional t-distribution, wherein eachmulti-dimensional t-distribution has a different degree-of-freedomparameter and the groups in between each other still show dependencythrough correlation matrix and have tail dependency. Having built such agrouped t-copula model, this model can be calibrated using historicaldata in the same way as a t-copula model is calibrated with theexception that a maximum likelihood estimation of the multiple degreesof freedom parameters is performed separately on each of the multiplerisk factor groups. Simulation is afterwards also performed in the sameway as when using the t-copula model, and the same types of risk measurevalues are obtained.

It was empirically found, that when using the grouped t-copulas theresulting risk measure values are different from the one obtained byusing the usual ungrouped t-copulas. It was therefore observed that thenew system and method is better able to capture the risk in a large setof risk factors.

While the present invention will hereinafter be described in connectionwith a preferred embodiment and method of use, it will be understoodthat it is not intended to limit the invention to this embodiment.Instead, it is intended to cover all alternatives, modifications andequivalents as may be included within the spirit and scope of thepresent invention as defined in the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be more clearly understood with reference tothe following detailed description of preferred embodiments, taken inconjunction with the accompanying drawings, in which

FIG. 1 illustrates the inventive system S and its input data and

FIG. 2 illustrates the inventive system S being used for a multiple ofportfolios.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

In the system and method according to the invention, Z˜N_(d)(0,ρ), whereρ is an arbitrary linear correlation matrix, is independent of U, arandom variable uniformly distributed on [0,1]. Furthermore, G_(y)denotes the distribution function of √{square root over (ν/x_(ν) ²)}.According to the invention, a partition {l , . . . , d} into m subsetsof sizes say s₁, . . . , s_(m) is made, wherein m is different from 1.R_(k)=G_(ν) _(k) ⁻¹(U) for k=1, . . . , m. IfY=(R ₁ Z ₁ , . . . , R ₁ Z _(s) ₁ ,R ₂ Z _(s) ₁ ₊₁ , . . . ,R ₂ Z _(s) ₁_(+s) ₂ , . . . ,R _(m) Z _(d))′,  (7)then the random vector (Y₁, . . . . Y_(s) ₁ )′ has an s₁-dimensionalt-distribution with ν₁ degrees of freedom and, for k=1, . . . , m−1,(Y_(s) ₁ _(+. . . +s) _(k) ₊₁, . . . , Y_(s) ₁ _(+ . . . +s) _(k+1) )′has an s_(k)+1-dimensional t-distribution with ν_(k)+1 degrees offreedom. Finally, F_(k) denotes the distribution function of Y_(k) andH₁, . . . , H_(d) are some arbitrary continuous distribution functions.X=(H ₁ ⁻¹(F ₁(Y ₁)), . . . ,H _(H) ⁻¹(F _(J)(Y _(J))))′is therefore a generalisation of the meta-t model which allows differentsubsets of the components to have different degrees of freedomparameters ν_(m). Its copula model will hereinafter be called groupedt-copula model.

An example is given for d=4, i.e. for four risk factors, wherein eachrisk factor belongs to an equity. In this example, two of these fourequities are from US, two of them are from Switzerland. This example isnot representative since the models usually comprise a multiple of riskfactors:

The 4 risk factors are described by 4 random variables that aredependent among each other. According to the invention, the 4 randomvariables are divided in groups. Here, we choose to divide them bycountry: i.e. we receive two groups of two risk factors, wherein eachgroup represents a country.

In order be able to run a simulation according to the invention, a 4drandom vector Y═(Y₁,Y₂,Y₃,Y₄) with grouped t-copula dependency among the4 components is needed.

By using standard techniques, the linear correlation ρ for the 4 riskfactors is determined. Then, two additional parameters ν₁ and ν₂ areintroduced which will take into account tail dependency and furtherallows a different tail dependency in group 1 and 2. For example, ν₁=4,which describes a high tail dependency, and ν₂=25, which describes a lowtail dependency.

Z=(Z₁,Z₂,Z₃,Z₄), G₁ and G₂ are random variables having the followingdistribution:Z˜N₄(0,ρ)G₁˜√{square root over (ν₁/χ_(ν) ₁ ²)}G₂˜√{square root over (ν₂/χ_(ν) ₂ ²)}with x_(ν) ² the usual Chi Square distribution.

U is independent of Z and uniformly distributed on [0,1]. The two newrandom variables R₁ and R₂ with distribution areR ₁ =G ₁ ⁻¹(U) R ₂ =G ₂ ⁻¹(U)

Finally construct

$Y = \begin{pmatrix}{R_{1}Z_{1}} \\{R_{1}Z_{2}} \\{R_{2}Z_{3}} \\{R_{2}Z_{4}}\end{pmatrix}$

This is by definition a random vector having a grouped t-copula.

The grouped t-copula can be written down in a form similar to (3).However, because of the multidimensional case, the expression is quitecomplex and it is therefore not given explicitly. A person skilled inthe art will know how to write this expression. We believe, that theproperties of the grouped t-copula is best understood from (7) and theabove stochastic representation. Moreover, for calibration of thegrouped t-copula model with historical data and for simulation using thecalibrated grouped t-copula model there is no need for an explicitcopula expression, as can be seen below: The simulation from theinventive grouped t-copula is no more difficult than simulation from at-copula. The simulation comprises the following steps:

(i) Draw independently a random variate Z from the d-dimensional normaldistribution with zero mean, unit variances and linear correlationmatrix ρ, and a random variate U from the uniform distribution on [0,1].

(ii) Obtain R₁, . . . ,R_(m) by setting R_(k)=G_(ν) _(k) ⁻¹(U) for k=1,. . . , m. By (7) we obtain a random variate (Y₁, . . . Y_(d))′ from thegrouped t-distribution.

(iii) Finally,(t_(ν) ₁ (Y₁, . . . ,t_(ν) ₁ (Y_(s) ₁ ),t_(ν) ₂ (Y_(s) ₁ ₊₁), . . .,t_(ν) ₂(Y_(s) ₁ _(+s) ₂ ), . . . ,t_(ν) _(m) (Y_(d))′is a random variate from the grouped t-copula.

The calibration of this model is identical to that of the meta-tdistribution except that the maximum likelihood (ML) estimation of the mdegrees of freedom parameters ν_(k) has to be performed separately oneach of the m risk factor groups. The key point is that theapproximation

$\begin{matrix}{{{\tau\left( {X_{i},X_{j}} \right)} \approx {\tau\left( {Z_{i\;,}Z_{j}} \right)}} = {\frac{2}{\pi}{\arcsin\left( \rho_{ij} \right)}}} & (8)\end{matrix}$is very accurate. Again, the eigenvalue method may have to be applied toassure positive definiteness.

In the following, a specific example is given to render the inventivemethod to be more clear:

We consider an internationally diversified credit portfolio with Kcounterparties. It is assumed that the systematic risk of eachcounterparty is adequately described by a set of risk factors, which are92 country/industry equity indices as shown in Table 1. These riskfactors are divided into 8 groups defined by country. The divisionaccording to the countries is only one way to form groups. Otherdivisions, such as divisions according to industrial sectors, arepossible too.

According to the invention, this grouped t-copula is used to describethe dependence structure of the risk factors, and complete the model byspecifying normally distributed marginals. The marginals for monthlyreturns are assumed to be normally distributed. Other distributions arepossible as well.

For our example, we consider a single counterparty k and take a timehorizon T=1 month. I_(k) is the state variable for counterparty k attime horizon T. In this example, we consider only default events and notthe impact of upgrades or downgrades on the credit quality. Therefore,we assume that I_(k) takes values in {0,1}: the value 0 represents thedefault state, the value 1 is the non-default state.

Y_(k) is a random variable with continuous distribution functionF _(k)(x)=P└Y _(k) ≦x┘.d_(k)εR and setI _(k)=0

Y _(k) ≦d _(k)  (9)

The parameter d_(k) is called the default threshold and (Y_(k),d_(k)) isthe latent variable model for I_(k). The following interpretation is puton Y_(k). Let A_(t) ^(k) be the asset value of counterparty k at time t.We put

${Y_{k} = {\log\left( \frac{A_{T}^{k}}{A_{O}^{k}} \right)}},$i.e. Y_(k) is defined as the asset value monthly log return. A defaultoccurs if the asset value log return falls below the threshold d_(k).

We find parameters a_(k) and λ_(k)ε[0,1] such thatY _(k)=√{square root over (λ_(k))}a′ _(k) X+√{square root over(1−λ_(k))}s _(k)ε_(k),  (10)where X is the vector of monthly risk factor log returns, with a groupedt-copula and normally distributed marginals, E[X]=0 and ε_(κ)˜N(0,1),independent of X. The model (10) says that asset value monthly logreturn can be linked to the risk factors by a′_(k) X, which gives thesystematic component of the risk and same additional independentidiosyncratic component ε_(K). The parameter λ_(k) is the coefficient ofdetermination for the systematic risk (how much of the variance can beexplained by the risk factors) ands _(k) ²=Var(Y _(k))=a′ _(k)Cov(X)a _(k).

Let π_(k) be the unconditional probability of default for counterpartyk, i.e. π_(k)=F_(k)(d_(k))·π_(k) is assumed to be given from someinternal or external rating system or other procedures. The conditionalprobability of default for counterparty k given the risk factors X canbe written as

${{Q_{k}(X)} = {{P\left\lbrack {{Y_{k} \leq d_{k}}❘X} \right\rbrack} = {\Phi\left( \frac{{F_{k}^{- 1}\left( \pi_{k} \right)} - {\sqrt{\lambda_{k}}a_{k}^{\prime}X}}{\sqrt{1 - \lambda_{k}}\sqrt{a_{k}^{\prime}{{Cov}(X)}{a_{k}}_{\;}}} \right)}}},$where Φ denotes the standard normal cumulative distribution function. Inthe classical model, Y_(k) is normally distributed and thus theπ_(k)-quantile F_(k) ⁻¹(π_(k)) can be easily computed. Here, thedistribution function of Y_(k) is unknown: F_(k) ⁻¹(π_(k)) is replacedby the empirical quantile estimate {circumflex over (F)}_(k) ⁻¹(π_(k)).Consequently, the estimated conditional probability of default{circumflex over (Q)}_(k)(X) is obtained by replacing F_(k) ⁻¹(π_(k)) by{circumflex over (F)}_(k) ⁻¹(π_(k)) in the equation for {circumflex over(Q)}_(k)(X).

This default model described by equations (9) and (10) is applied toeach single counterparty in the credit portfolio. The counterpartiesdefaults are handled as being conditionally independent, given the riskfactors X, i.e. the ε_(κ)'s are independent.

For each scenario X for the risk factors, counterparty defaults aresimulated from independent Bernoulli-mixture distributions withparameters {circumflex over (Q)}_(k)(X). Naturally, one could alsosimulate Y_(k) using equation (8) so that a default occurs if thesimulated Y_(k) is smaller than the estimated default threshold{circumflex over (F)}_(k) ⁻¹(π_(k)) The advantage of using theBernoulli-mixture model is that it can be easily extended to aBinomial-mixture model for a sub-portfolio of homogeneouscounterparties.

I_(k)(X)ε{0,1} is the conditional default indicator for counterpartyk,E_(k) is the corresponding exposure and l_(k) is the loss givendefault. Then L(X)=Σ_(k=1) ^(K)I_(k)(X)l_(k)E_(k) gives the total creditloss under scenario X.

In summary, the credit loss distribution is obtained by a three stageprocedure:

(i) Simulation of the monthly risk factor log returns X from a groupedt-copula with normal marginals;

(ii) For each counterparty k, simulation of the conditional defaultindicator I_(k)(X) ε{0,1} from a Bernoulli-mixture model withconditional default probability {circumflex over (Q)}_(k)(X);

(iii) Estimation of the credit loss distribution over a large set ofscenarios for X, by integrating exposures and loss given default in theloss function L(X).

In this example, we calibrate the grouped t-copula and the normallydistributed marginals using monthly risk factor log returns from 1992 to2002 (hence 120 observations) obtained from DataStream™. Table 1 showsthe estimated degrees of freedom parameters for various subsets of riskfactors and the overall estimated degrees of freedom parameter. Becauseof the difference between the various subset degrees of freedomparameters a grouped t-copula is more appropriate for describing thedependence structure.

Set Number of risk factors ν AUS Indices  9 15 CAN Indices 14 24 CHIndices  4 19 FRA Indices  5 67 GER Indices 10 65 JPN Indices 15 14 UKIndices 15 17 US Indices 20 21 All 92 29Table 1: Estimated degrees of freedom ν for various sets of riskfactors. The country equities indices are for major industrial sectors.

The credit portfolio contains K=200 counterparties with the sameunconditional default probability π=1%. Each counterparty is assigned toa country so that there are 25 from each country. The weights a_(k) andλ_(k) (k=1, . . . , 200) are generated as follows. For λ_(k) we randomlychoose values between 20% and 60%, which are common in credit modelling.Each counterparty is then described by two different risk factors(labelled i₁ and i₂) from the country to which it has been assigned, andthe value of a_(k) ^(i) ¹ (and hence also that of a_(k) ^(i) ² ) aredrawn from a uniform distribution on (0,1) such that a_(k) ¹ +a_(k) ^(i)² =1. Moreover, each counterparty has a total exposure of 1000 CHF andthe loss given default is assumed to be uniformly distributed on [0,1].

500'000 simulations were performed using a 92-dimensional Sobolsequence. The simulated default frequencies were all in the range0.97%-1.03% and the expected value of the portfolio loss distributionwas estimated with less than 0.2% error.

In the following the results are presented by making a comparisonbetween our new system incorporating the grouped t-copula (with degreesof freedom parameters shown in Table 1) and (1) a model with a t-copula(with 29 degrees of freedom) and (2) a model with a Gaussian copula. TheGaussian model was taken as baseline and the differences in the riskmeasures were expressed as percentages. In Table 2 the various riskmeasures for the total credit loss distribution are shown. Taking thetail dependence into account with the t-copula gives an increasedassessment of the risk. By introducing the grouped t-copula, even largerrisk measures were received. The 99%-shortfall is in this case more than10% larger than in the normal case.

Measure T₂₉ grouped-t Max. value 29.7% 41.4% Std dev. 4.2% 5.3% 95%quantile 1.1% 1.7% 99% quantile 4.3% 6.0% 95% shortfall 3.4% 4.8% 99%shortfall 8.9% 10.8%Table 2: Risk measures of the sample portfolio using a t₂₉-copula or agrouped t-copula to model the dependence among the 92 risk factors. Thevalues shown are the percentage deviations from those obtained with thenormal copula.

The above described inventive method is preferably performed by use of adata processing system using at least one computer. This system S asshown in FIG. 1 comprises

-   -   modelling and calibration means Mod/Cal, this means Mod/Cal        comprising a program which describes d risk factors as random        variables X₁ to X_(d), the random variables being related to        each other by a correlation matrix ρ, wherein this means        -   form m groups of the random variables X₁ to X_(d),        -   describe the random variables X₁ to X_(d) as a d-dimensional            random vector X, thereby forming m subvectors, each            subvector consisting of one group of the random variables X₁            to X_(d)        -   describe the dependencies of the risk factors as the            implicit copula of a d-dimensional random vector Y, this            random vector Y consisting of m subvectors Y_(k) (k=1 to m,            m≠1), wherein each subvector Y_(k) has a t-distribution with            a parameter ν_(k), this parameter describing a degree of            freedom, and wherein its copula being a t-copula; wherein d,            m and k are natural positive numbers;    -   input means (a, b, c)        -   for entering or choosing calibration data for obtaining by            using the modelling and calibration means values for the            ν_(k) degrees of freedom parameters for each of the m            subvectors Y_(k) separately and for obtaining values for the            correlation matrix ρ for all the random variables X₁ to            X_(d),        -   for entering or choosing at least one risk mapping function            L(X), in particular a profit and loss function, and        -   for entering portfolio data of the portfolio to be analysed;    -   simulation means SIM for simulating realisation of the d risk        factors by using the calibrated correlation matrix ρ, the        calibrated values ν_(k) of the degrees of freedom parameters,        the risk mapping function L(X) and the portfolio data of the        portfolio and    -   output means for showing the output data of the simulation in        the form of a risk measure or a price.

In a preferred embodiment, the system comprises at least three inputlevels:

-   -   a first level comprising a first input means (a) for entering or        choosing the calibration data, wherein these data are used by        the modelling and calibration means Mod/Cal;    -   a second level comprising a second input means (b) for entering        or choosing at least one risk mapping function L(X), wherein        this function is preferably handled by a risk mapping means RM;        this risk mapping means RM can be a stand alone means or being        part of the simulation means SIM; and    -   a third level with third input means (c) for entering the        specific portfolio data being used by the simulation means SIM.

The calibration data are all the data needed for calibrating the model.Usually, they comprise historical data, marginals and informationconcerning the groups to be formed, i.e. a maximum number of groups andthe information in view of which aspects or criteria the groups areformed. The risk mapping function is preferably a profit and lossfunction and it depends of the general type of portfolio to be analysed.The portfolio data depend on the specific portfolio to be analysed andcan change daily.

The calibration is performed periodically, for example once a year, withupdated data. The risk mapping function must only be changed when a newgeneral type of portfolio is entered. The portfolio data are enteredmore often, i.e. each time, when an updated risk measure or a new prizeshall be obtained. Depending on the kind of business and the kind ofportfolio, this is usually done daily or at least once a week.

The system can be used by different users, which allows the users tohave different levels of mathematical understanding. The modelling andcalibration steps are usually performed by a first person, this personusually having a fundamental mathematical background. The risk mappingstep is performed by a second person, who is usually a well trainedsenior risk analyst and has preferably some sort of mathematicalbackground. The simulation is done by a risk analyst being responsiblefor the portfolio.

As can be seen in FIG. 2, the system also allows to perform simulationwith different types of portfolios and with different portfolios withinthe same type, thereby using the same modelling and calibration meansMod/Cal. The calibration data then comprise information about all theportfolios to be handled. For example, when a first portfolio comprises50′ equities of 10 countries and a second portfolio comprises 70equities of 20 countries, 30 of the equities and 5 of the countriesbeing the same as in the first portfolio, the calibration data consider90 different kinds of equities and can define 25 groups of differentcountries. For each type of portfolio, there exists a separate riskmapping means RM for entering the specific risk mapping function L(X).For each kind of portfolio, a separate simulation SIM can be performed.

In a preferred embodiment of the system S, the system comprises a datastorage for storing the historical data. However, it is also possible tostore the historical data on other means and to transfer it into thesystem when performing the calibration. It is also possible to use adifferent computer or subsystem for the calibration and the simulation,transferring the data from the calibration computer or subsystem to thesimulation computer or subsystem. The inventive system may then comprisestoring means for storing the calibrated correlation matrix ρ and thecalibrated parameters ν_(k) describing the degrees of freedom. Likethis, simulations can be run on different computers at the same time.

Preferably, the inventive system further comprises input means forgrouping the d interdependent risk factors manually or for choosingmanually a grouping from a range of several kind of groupings. Thisenables a user to group the risk factors according to the countries oraccording to the industrial sectors or other criteria.

As can be seen, grouping the t-copulas enables to model large sets ofrisk factors of different classes. This grouped t-copula has theproperty that the random variables within each group have a t-copulawith possibly different degrees of freedom parameters in the differentgroups. This gives a more flexible overall dependence structure moresuitable for large sets of risk factors. When calibrated to a historicalrisk factor data set, the system allows to more accurately model thetail dependence present in the data than the popular Gaussian andt-copulas.

1. A computerized data processing system including a computer forperforming risk analysis of a portfolio, the system comprising: amodeling and calibration unit implemented on the computer, configured todescribe d risk factors as random variables X₁ to X_(d), the randomvariables being related to each other by a correlation matrix ρ, to formm groups of the random variables X₁ to X_(d), to describe the randomvariables X₁ to X_(d) as a d-dimensional random vector X, forming msubvectors, each subvector including one group of the random variablesX₁ to X_(d), and to describe dependencies of the risk factors as theimplicit copula of a d-dimensional random vector Y, the random vector Yincluding m subvectors Y_(k) (k=1 to m, m≠1, d, m, and k are naturalpositive numbers), each subvector Y_(k) has a t-distribution with aparameter ν_(k) describing a degree of freedom, and a copula of eachsubvector Y_(k) is a t-copula; an input unit implemented on thecomputer, configured to enter or choose calibration data to obtain, byusing the modeling and calibration unit, values for the ν_(k) describingthe degree of freedom for each of the m subvectors Y_(k) separately andto obtain values for a correlation matrix ρ for the random variables X₁to X_(d), to enter or choose at least one risk mapping function L(X),and to enter portfolio data of the portfolio to be analyzed; asimulation unit implemented on the computer, configured to simulatingrealization of the d risk factors by using the correlation matrix ρ, theparameters ν_(k) describing the degrees of freedom, the at least onerisk mapping function L(X), and portfolio data of the portfolio; and anoutput unit implemented on the computer, configured to generate outputdata resulting form the simulation unit in a form of at least one of arisk measure or a price.
 2. The computerized system according to claim1, wherein the input unit is further configured to enter at least threeinput levels, a first level including a first input unit configured toenter or choose the calibration data, a second level including a secondinput unit configured to enter or choose the least one risk mappingfunction L(X), and a third level with third input unit configured toenter the portfolio data.
 3. The computerized system according to claim1, further comprising a data storage unit configured to store thehistorical data.
 4. The computerized system according to claim 1,wherein the input unit is further configured to group the d risk factorsmanually or configured to manually choose a grouping from a range ofseveral kind of groupings.
 5. The computerized system according to claim1, further comprising: a storing unit in the computer configured tostore the correlation matrix ρ and the parameters ν_(k) describing thedegrees of freedom.
 6. A computer implemented method for performing riskanalysis of a portfolio on a computer by combining d interdependent riskfactors to determine a risk measure or a price, the method comprising: astep of building a model with the computer by describing the d riskfactors as random variables X₁ to X_(d) being related to each other by acorrelation matrix ρ, forming m groups of the random variables X₁ toX_(d), describing the random variables X₁ to X_(d) as a d-dimensionalrandom vector X to form m subvectors, each subvector including one groupof the random variables X₁ to X_(d), and describing dependencies of thed risk factors as an implicit copula of a d-dimensional random vector Y,the random vector Y including m subvectors Y_(k) (k=1 to m), whereineach subvector Y_(k) has a t-distribution with unknown values ofparameters V_(k), the values of parameters V_(k) describing degrees offreedom, and wherein a copula of the values of parameters V_(k) being at-copula; a step of calibrating the model with the computer of said stepof building by using historical data to obtain values for the parametersV_(k), describing degrees of freedom for each of the m subvectors Y_(k)separately, and to obtain values for the correlation matrix ρ for allthe random variables X₁ to X_(d), wherein d, m and k are naturalpositive numbers; a step of simulating realization of the d risk factorswith the computer by using the calibrated model of the step ofcalibrating; and a step of generating output data with the computer inform of at least one of a risk measure or price based on said step ofsimulating.
 7. The method according to claim 6, wherein d is equal orgreater than 4 and each group of the random variables X₁ to X_(d)includes at least two random variables.
 8. The method according to claim6, wherein in said step of simulating on the computer, Kendal's tau isestimated for every group of subvectors Y_(k).
 9. The method accordingto claim 6, wherein said step of calibrating the model with the computerfurther comprises a maximum likelihood estimation of the m degrees offreedom.
 10. The method according to claim 6, wherein the randomvariables X₁ to X_(d) are grouped in m groups according to predefinedaspects, countries, or industrial sectors.
 11. The method according toclaim 6, wherein said step of building a module with the computerfurther comprises building the model by specifying normally distributedmarginals.